Average Error: 38.6 → 0.5
Time: 10.0s
Precision: 64
Internal Precision: 1344
$e^{x} - 1$
$\begin{array}{l} \mathbf{if}\;e^{x} - 1 \le -1.4757203902199797 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{x + x} - 1}{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right) + x\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (- (exp x) 1) < -1.4757203902199797e-11

1. Initial program 0.6

$e^{x} - 1$
2. Using strategy rm
3. Applied flip--0.6

$\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}$
4. Applied simplify0.5

$\leadsto \frac{\color{blue}{e^{x + x} - 1}}{e^{x} + 1}$

## if -1.4757203902199797e-11 < (- (exp x) 1)

1. Initial program 59.0

$e^{x} - 1$
2. Taylor expanded around 0 0.5

$\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2} + x\right)}$
3. Applied simplify0.5

$\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right) + x}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 10.0s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "exp(x)-1"
(- (exp x) 1))